You know those big, majestic looking whitetail deer you see in a variety of places? You would very rarely see those. Instead, because of overpopulation, you'd see smaller bodied deer. Your car insurance would also go up noticeably, because greater deer populations means more claims from people hitting them - more insurance claims means across the board rate increases.
All those wetlands in the deep south that so many creatures rely on for habitat? Kiss those goodbye, coastal damage from nutria is already destroying marshes in Louisiana, and we can't exterminate enough of them to get ahead of it. Imagine what they will do when completely unchecked.
Better keep your pets inside. Coyotes are one of the most adaptable creatures in North America, and are doing a pretty good job of expanding into populated areas. Small dogs, house cats, chickens, etc., are all far easier targets as a food source than anything they encounter in the wild.
Feral pigs are already causing more than a billion dollars of damage to agriculture yearly. Remove hunting from our toolkit for their removal, and what happens? More tax dollars get spent on government programs to control them. More tax dollars being spent means more tax dollars being collected from you and me.
These are just a few examples that I know of off top of my head.